Preheat the oven to 350°F. Line a baking sheet with parchment paper.
Beat the egg.
Finely chop the onion.
Mince the oregano (if using fresh oregano).
Thoroughly mix the chicken, egg, onion, garlic, oregano, salt, and pepper in a large bowl. Roll into 15 to 20 meatballs, each about the size of a golf ball.
Melt the cooking fat in a large skillet over medium-high heat. When the fat is hot, add the meatballs (depending on the size of your pan, you may have to do this in batches). Cook for about 30 seconds per side, turning to prevent burning, until browned all over, about 5 minutes. Reduce the heat and add more cooking fat if the pan begins to smoke.
Transfer the meatballs to the prepared baking sheet. Transfer to the oven to finish cooking for 8-10 minutes, until the internal temperature reaches 160°F. Let the meatballs rest for 5 minutes and serve.